Anatomy of Pearl and Abrikosov Vortices: A Unified Framework for Core and Screening Physics (Part I)

We present a systematic study of Abrikosov vortex lines in a bulk superconductor and Pearl vortices in a thin films, encompassing the general case of finite coherence length ξ and finite Pearl length Λ, as well as all limiting cases. We solve the radial Ginzburg-Landau equations using complementary analytic and numerical methods – including short- and long-distance series, shooting, relaxation, variational schemes, and linear algebra methods – to obtain the superfluid amplitude, vector potential, current density, magnetic field, and free energy. We show the vortex free energy can be expressed as F_v∝ln[(Λ_e/ξ_e)η(Λ/ξ)], where the short-range cutoff ξ_e≈0.966ξ characterizes the vortex core size and the long-range cutoff Λ_e≈1.123Λ captures magnetostatic screening. In the limit Λ/ξ→∞, the function η→1, clearly separating core and screening effects. Finally, we derive approximations for the Gibbs free energy of a vortex pinned by a circular hole in the limit r_h<<ξ<<Λ.